This doesn't compile and it also doesn't if I remove the signatures from the declared class. If I only had one index signature in the interface, I could remove it from the class declaration and all is well. Is this something that can be supported?

I just ran into this as well. I needed to define an interface containing a property with two different possible types and this isn't currently allowed. Given that this is a fairly common scenario in JavaScript I think the team should consider
supporting this.